As Power of Attorney, am I responsible if my mother injures her caregiver?

Is the caregiver thru an agency or private-pay?
If caregiver is thru an agency make the home health agency aware of possible aggressive behavior. In turn they will relate that to the caregiver. I worked for home health agencies and was told up front of patients behaviors. So that would eliminate you from being responsible. Home health agencies carry insurance on all caregivers in case of injury.
If caregiver is private-pay, please make sure they know as well. I am sure that if you have a private-pay caregiver you have some sort of care plan or documentation maybe they signed off on. Amend that and review and make caregiver sign form. That makes them written and orally informed of possible behavior problems. With that said, that should remove any responsiblity from you. However, I would suggest talking to whomever you or mom has homeowners insurance or renters insurance thru and find out if someone gets hurt on your property is it covered. That would proctect you from a lawsuit if it gets that out of control.

Oh my goodness, Well, does your mother have dementia? Does she have anger issues? Does she become violent toward you and others? If the answer is yes to any of these questions I would get her to a Dr and get a diagnoses. This will possibly lead to placing your mother where she can't injure others but better than a law suit which could happen if the caregiver sues the family for liable.

You didn't really go into enough detail for anyone to give you a GOOD answer to your question. Good Luck and I hope this helps.